Intrinsic Linking and Knotting Are Arbitrarily Complex in Directed Graphs

INTRINSIC LINKING AND KNOTTING ARE ARBITRARILY COMPLEX IN DIRECTED GRAPHS

THOMAS W. MATTMAN, RAMIN NAIMI, AND BENJAMIN PAGANO

Abstract. Fleming and Foisy [4] recently proved the existence of a digraph whose every embedding contains a 4-component link, and left open the possibility that a directed graph with an intrinsic n-component link might exist. We show that, indeed, this is the case. In fact, much as Flapan, Mellor, and Naimi [2] show for graphs, knotting and linking are arbitrarily complex in directed graphs. Specifically, we prove the analog for digraphs of the main theorem of their paper: for any n and α, every embedding of a suffi- ciently large complete digraph in R3 contains an oriented link with components Q1,...,Qn such that, for every i 6= j, |lk(Qi,Qj)| ≥ α and |a2(Qi)| ≥ α, where a2(Qi) denotes the second coefficient of the Conway polynomial of Qi.

1. Introduction Fleming and Foisy [4] recently proved the existence of a digraph whose every embedding contains a 4-component link, and left open the possibility that a directed graph with an intrinsic n-component link might exist. We show that, indeed, this is the case. In fact, much as Flapan, Mellor, and Naimi [2] show for graphs, knotting and linking are arbitrarily complex in directed graphs. Speciﬁcally, we prove the analog for digraphs of the two main theorems of their paper. Before stating the results, we introduce some notation. For graph G, the symmetric digraph DG is obtained by replacing each edge vivj with two directed edges, vivj and vjvi. For cycle C in a digraph, let p1, . . . , pδ arXiv:1901.01212v1 [math.GT] 4 Jan 2019 in C be maximal consistently directed paths (no pi is a subpath of a longer consistently directed path in C). Then δ is the directionality of C. Thus, a consistently directed cycle is 1–directional. Following [2], we deﬁne the linking pattern of a link of n components, L1,...,Ln, as the weighted graph on vertices v1, . . . , vn where |lk(Li,Lj)| (if nonzero)

Date: January 7, 2019. 2010 Mathematics Subject Classiﬁcation. Primary 05C10, Secondary 57M15, 57M25, 05C20, 05C35 . Key words and phrases. intrinsically knotted graph, intrinsically linked graph, directed graph, spatial graph. 1 2 THOMAS W. MATTMAN, RAMIN NAIMI, AND BENJAMIN PAGANO is the weight of vivj. When the linking number is zero, there is no edge. The mod 2 linking pattern instead carries the weights ω(Li,Lj) = lk(L1,Lj) mod 2.

Theorem 1. Let λ, δ ∈ N with δ even or 1. For every n ∈ N, there is a digraph DG such that every embedding of DG in R3 contains a link whose weighted linking pattern is Kn with every weight at least λ and every component δ-directional.

Theorem 2. For every n, α ∈ N, there is a complete digraph DKr such 3 that every embedding of DKr in R contains a link with 1-directional components Q1,...,Qn such that for every i 6= j, |lk(Qi,Qj)| ≥ α and |a2(Qi)| ≥ α. Not just the statements of our theorem are similar to [2], but the proofs as well. We prove Theorem 1 in the next section and Theorem 2 in Section 3.

2. Intrinsic Linking In this section we prove Theorem 1. After a couple of introductory lemmas, we follow the same path as in [2]. Throughout this paper, indices are cyclic; e.g., given say xi with 1 ≤ i ≤ n (or 0 ≤ i ≤ n), xn+1 is to be understood as x1 (x0); and, more generally, for i > n, xi is to be understood as xi−n (xi−n−1).

Lemma 1. Every spatial digraph DK6m contains at least m pairwise disjoint 2-component links, all with odd linking numbers, such that all their components are 2-directional.

Proof. Take an undirected graph K6 with vertices v1, . . . , v6 and form the digraph DG by orienting the edges such that vivj is directed from vi to vj when i < j. Note that DG contains no 1-directional cycles. Additionally, notice that all 3-cycles in DG must be 2-directional. As argued in [1, 3], any embedding of K6 contains a pair of 3-cycles with odd linking number. Then, any embedding of DG must contain a link with 2-directional components. As DG is a subgraph of DK6 and DK6m contains m distinct copies of DK6, every embedding of DK6m in R3 contains m pairwise disjoint links with odd linking number and 2-directional components.

Lemma 2. Let M be an m × n matrix with entries in Z2 where every column of M contains at least one 1. For every n and m, there exists n a vector v ∈ row(M) for which over 2 of the entries are 1’s. LINKING AND KNOTTING IN DIGRAPHS 3

Proof. Let kvk denote the number of 1’s in a vector v. We proceed by induction on n. If n = 1, the statement is obvious. Fix m > 0 and assume that for all 1 ≤ l < n, the statement holds for every m × l matrix. Let m be an m × n matrix and suppose v0 is a vector in the row space that maximizes kv0k. For a contradiction, n assume that kv0k = k ≤ 2 . Without loss of generality, the ﬁrst k entries in v0 are 1, and the rest are all 0. Divide M into the two matrices ML and MR where the m×k matrix ML is the ﬁrst k columns of M and the m × n − k MR is the remaining n − k columns of M. Similarly, split vectors x ∈ row(M) into vectors xL and xR of lengths k and n − k respectively. For every x, kxLk ≥ kxRk, because otherwise kv0 + xk > kv0k, contradicting the maximality of kv0k. By induction, n−k n−k there is an x such that kxRk > 2 . This would imply kxLk > 2 , and kxk > n − k ≥ k = kv0|, a contradiction.

Lemma 3. Suppose a spatial digraph DKp contains links with components J1, ··· ,J2n and X1, ··· X2n such that Ji is 2-directional and ω(Ji,Xi) = 1 for every i ≤ n. Then DKp contains a 1-directional cycle Z in DKp with vertices on J1 ∪ · · · ∪ J2n such that for some I ⊆ {1, ··· , 2n} with |I| ≥ n/2, ω(Z,Xi) = 1 for all i ∈ I. Further- more, for every δ ≥ 1, Z can be chosen to be 2δ-directional if DKp contains at least 2δ − 2 vertices disjoint from all Ji and Xi.

Z J J1 J2 n

u w u1 w1 u2 w2 n n

X1 X2 Xn Xi Xi Xi 1 2 n 2

Figure 1. Illustration of Lemma 3

Proof. Since each Ji is 2-directional, it has exactly two vertices where “direction changes” on Ji, i.e., the two edges at each of these vertices are directed either both toward or both away from that vertex. We label these two vertices ui and wi, so that both paths on Ji between ui and wi are directed from ui toward wi. Let qi be one of these two 4 THOMAS W. MATTMAN, RAMIN NAIMI, AND BENJAMIN PAGANO directed paths; let wiui+1 denote the directed edge from wi to ui+1; and S S let C = qi ∪ wiui+1. Note that C is a 1-directional cycle. If we want the cycle Z in the conclusion of the lemma to be 2 directional, we make C 2-directional by replacing the edge w2nu1 in C with the edge u1w2n. And if we want Z to be 2δ directional with δ ≥ 2, we replace the edge w2nu1 in C with a path between w2n and u1 that goes through the 2δ − 2 extra vertices given in the hypothesis of the lemma, such that the path changes direction at each of those vertices and also at w2n and u1. If ω(C,Xi) = 1 for at least n/2 of the Xj’s, we let Z = C, and we are done. Otherwise, we construct Z as follows. Let M be the matrix with entries Mij = ω(Ji,Xj). Since Mii = 1 for all i, by Lemma 2, there exist rows i1, ··· , ik of M whose sum contains greater than n 1’s. Let

Z = C∇Ji1 · · · ∇Jik . Since C links fewer than n/2 of the Xj’s, while the sum of the rows i1, ··· , ik contains greater than n 1’s, it follows that Z links at least n/2 of the Xj’s, as desired. Recall from [2] that a generalized mod 2 keyring link is one whose mod 2 linking pattern includes an n-star. Proposition 1. Let n, δ, and ∈ N with each of δ and either even or 1. There is a digraph DG such that every embedding of DG in R3 contains a link whose mod 2 linking pattern contains the complete bipartite graph Kn,n, where every component of the ﬁrst partition is δ- directional and every component of the second partition is -directional. Proof. The argument largely follows the proof of the corresponding proposition in [2], and we begin by summarizing their approach. First observe that for any given m, there’s a p such that every embedding of Kp contains m disjoint mod 2 generalized keyrings, each having n keys. Let X1,...,Xm denote the rings. Apply (the analogue of) Lemma 3 n times to construct n cycles Z1,...,Zn and index set In of size at least n so that, for every i ∈ In and every j ≤ n, ω(Zj,Xi) = 1. The mod 2 linking pattern of the Zj’s and Xi with i ∈ In then contains Kn,n. It remains to modify the argument to take directionality into ac- count. By combining Lemmas 1 and 3, we can ﬁnd an embedding of DKp with a desired number m of disjoint mod 2 generalized keyrings, such that each ring Xi is -directional. By applying Lemma 3 n times we again construct In of cardinality at least n and δ-directional rings Z1,...,Zn such that each ω(Zj,Xi) = 1. 3 Lemma 4. Let λ ∈ N. Let DKp be embedded in R such that it contains a link with two-directional components J1, ··· ,Jr, L1, ··· ,Lq, m X1, ··· ,Xm, Y1, ··· ,Yn, where r ≥ m(2λ + 1)2 , q ≥ (m + n)(2λ + LINKING AND KNOTTING IN DIGRAPHS 5

m n 1)3 2 , and for every i, j, α, β, lk(Ji,Xα) 6= 0 and lk(Lj,Yβ) 6= 0. Then DKp contains a 1-directional cycle Z with vertices on J1 ∪ · · · ∪ Jr ∪ L1 ∪ · · · ∪ Lq such that for every α and β, |lk(Z,Xα)| > λ and |lk(Z,Yβ)| > λ. Furthermore, for every δ ≥ 1, Z can be chosen to be 2δ-directional if DKp contains at leat 2δ − 2 vertices disjoint from all Ji,Xα,Lj,Yβ.

J1 L1 Z J L X X X 2 2 Y Y Y m 2 1 1 2 n Xm X2 X1 Y1 Y2 Yn

Jr Lq

Figure 2. Illustration of Lemma 4

Proof. The first step in the proof of the corresponding lemma in [2] involves discarding Ji’s so that we are left with only positive linking numbers between each Ji and each Xα. To do this, note that at least half the linking numbers lk(Ji,X1) have the same sign. If this sign is negative, we reverse the orientation of X1 so that they become positive. Of the Ji’s which have positive linking number with X1, at least half have the same signed linking number with X2; we repeat the process, r eventually finding a set of at least 2m ≥ m(2λ+1) Ji’s which each have positive linking number with every Xα. We will assume without loss of generality that we are left with J1, ··· ,Jm(2λ+1). q m The same process can be used to find a set of 2n ≥ (m+n)(2λ+1)3 Lj’s which each have positive linking number with every Yβ. For one final discard, we wish to throw out some of the remaining Lj’s so that for each α, the linking numbers lk(Lj,Xα) are either positive, negative, or zero for all j. For each α, at least a third of the linking numbers lk(Lj,Xα) falls into one of these three categories, so, after discarding the Lj’s in the two other categories, we retain at least a third of the Lj’s. This process leaves us in the end with at least (m+n)(2λ+1)3m 3m = (m + n)(2λ + 1) Lj’s. We will assume without loss of generality that we are left with L1, ··· ,L(m+n)(2λ+1). Next, we create a cycle C0 with vertices on J1, ··· ,Jm(2λ+1), and L1, ··· ,L(m+n)(2λ+1). For i ≤ m(2λ+1), let ui and wi be the vertices on Ji where direction changes, and for j ≤ (m + n)(2λ + 1), let um(2λ+1)+j 6 THOMAS W. MATTMAN, RAMIN NAIMI, AND BENJAMIN PAGANO and wm(2λ+1)+j be vertices on Lj where direction changes. For i ≤ m(2λ + 1), let qi be the path from ui to wi on Ji which is directed opposite the orientation of Ji, and for j ≤ (m+n)(2λ+1), let qm(2λ+1)+j be the path from um(2λ+1) + j to wm(2λ+1)+j on Lj which is directed opposite the orientation of Lj. Additionally, for k < (2m + n)(2λ + 1), let ek be the edge in DKp directed from wk to uk + 1. For k = (2m + n)(2λ + 1), let e(2m+n)(2λ+1) be the edge directed from w(2m+n)(2λ+1) to u1 if a one-directional cycle Z is desired. If a 2-directional cycle Z is desired instead, let e(2m+n)(2λ+1) be the edge directed from u1 to w(2m+n)(2λ+1). And if a 2δ-directional cycle Z, where δ ≥ 2, is desired, let e(2m+n)(2λ+1) be a path in DKp from w(2m+n)(2λ+1) to u1 that uses the 2δ−2 additional vertices given in the hypothesis of the lemma, such that direction changes at w(2m+n)(2λ+1), at u1, and at every additional vertex. Now, let C0 be the union of all ek and qk for 1 ≤ k ≤ (2m+n)(2λ+1); and let Cs = C0∇J1∇ · · · ∇Js, 1 ≤ s ≤ m(2λ + 1). We orient C0 in the same direction as the qk’s (i.e., on each arc Lj ∩ C0, Lj and C0 have opposite orientations), and the orientation of each Cs is induced by that of C0. This implies lk(Cs+1,Xα) > lk(Cs,Xα) for every s, including s = 0. Now, consider the matrix A with entries Aα,s = lk(Cs,Xα), where 1 ≤ α ≤ m, 0 ≤ s ≤ m(2λ + 1). Observe that the entries in each row of A are pairwise distinct; so in each row at most 2λ + 1 entries have magnitude less than or equal to λ. Since A has m rows, it contains at most m(2λ + 1) entries which have magnitude less than λ. On the other hand, A has m(2λ + 1) + 1 columns, which implies at least one of the columns has no entry less than or equal to λ. In other words, for some s, |lk(Cs,Xα)| > λ for every α. We let D0 denote this cycle Cs. Recall that for each α, lk(Xα,Lj) has the same sign (+, −, or 0) for every j. For each Xα, by reversing its orientation if necessary, we can assume its linking number with every Lj is non-negative. Note that this does not change the fact that |lk(D0,Xα)| > λ for every α. Now, let S be the set of all Yβ’s and all Xα’s which have positive linking number with every Lj. Thus, S contains all Yβ’s and some Xα’s — at most m + n cycles altogether. For 1 ≤ t ≤ (m + n)(2λ + 1), let Dt = D0∇L1∇ · · · ∇Lt . Then by a similar argument as above, there is some t such that |lk(Dt,A)| > λ for all A ∈ S. Let Z denote this cycle Dt. Observe that for each Xα 6∈ S, |lk(Z,Xα)| = |lk(D0,Xα)| > λ since lk(Xα,Lj) = 0. Thus Z is our desired cycle. Proof. (of Theorem 1) The proof is largely similar to the corresponding theorem in [2]. The main difference is we work with digraphs and must LINKING AND KNOTTING IN DIGRAPHS 7 pay attention to the directionality of cycles. For m, n ∈ N, let H(n, m) denote the complete (n + 2)-partite graph with two parts (P1 and P2) of size m and the remaining parts (Q1,...,Qn) being single vertices. By induction on n, for every n ≥ 0 and m ≥ 1, we will show there is a digraph DG such that every embedding of DG includes a link whose linking pattern contains H(n, m) with Qi to Qj edges of weight greater than λ and each cycle represented by a Qi vertex δ-directional. When n = 0, H(0, m) = Km,m. By Proposition 1, for every m, there is a digraph DG so that every embedding includes a link with linking pattern containing Km,m. Moreover, we can assume that all cycles in the link are 2-directional. For the inductive step, assume that, for some n ≥ 0 and every m ≥ 1, there is a digraph DG such that every embedding if DG in R3 includes a link with linking pattern containing H(n, m), where the weight of every Qi,Qj edge exceeds λ, cycles for vertices in P1 and P2 are 2-directional, and those for Qi vertices are δ-directional. Given m, let q = (2m + n)(2λ + 1)3m2m+n and s = m + q. In the graph H(n, s), label the s vertices in P1, X1,...,Xm,L1,...,Lq and those in P2, Y1,...,Ym,J1,...,Jq. For i ≤ n let Ym+i denote the vertex inQi. By induction, there is a DG that includes, in each embedding, a link L with linking pattern containing H(n, s) with the desired weights and directionalities. Without loss of generality, we assume DG is a 3 complete graph DKp. Fix an embedding of DKp in R . We will show that this embedding also contains a link whose weighted linking pattern contains H(n + 1, m) with the desired weights, and whose components possess the desired directionality. By abuse of notation, we denote the components of L in DKp by the name of the vertex that represents the component in the linking pattern. Applying Lemma 4 to the link in DKp with components J1,...,Jq,L1,...,Lq,X1,...,Xm, and Y1,...,Ym+n where r = q = m m+n (2m + n)(2λ + 1)3 2 , we find a δ-directional cycle Ym+n+1 where |lk(Ym+n+1,Xα)| > λ and |lk(Ym+n+1,Yβ)| > λ for every α ≤ n and β ≤ m + n. 0 Thus, DKp inludes a link L with components X1,...,Xm, and Y1,...,Ym+n+1, which can be partitioned into subsets corresponding 0 0 to the vertices of H(n + 1, m). Namely P1 is the Xi components, P2 0 0 are the first m Yi’s and the remaining Yi’s go, one each, to a Qi. In L , every component in one partition is linked with every component in 0 all other partitions, each component in a partition Qi is δ-directional, and for every i 6= j where i, j ≤ n + 1, |lk(Ym+i,Ym+j)| > λ. Thus, the weighted linking pattern of L0 contains H(n + 1, m), with the desired 8 THOMAS W. MATTMAN, RAMIN NAIMI, AND BENJAMIN PAGANO weights and directionality for every vertex and edge among partitions 0 0 Q1,...,Qn+1. Therefore, we have proven that for every n ≥ 0 and m ≥ 1, there is a digraph DG whose every embedding in R3 contains a link whose linking pattern contains H(n, m) where the weight of every edge between vertices Ym+i and Ym+j in Qi and Qj respectively is greater than λ, and the cycle represented by Ym+i is δ-directional for every i ≤ n. As Kn is a subgraph of H(n, m), we have shown that every embedding 3 of G in R contains a link whose linking pattern is Kn, the weight of every edge of Kn being greater than λ, and every cycle represented by a vertex in Kn being δ-directional. 3. Intrinsic knotting In this section, we prove Theorem 2. Even more than what has gone before, we follow closely the argument of [2]. We begin with two definitions from that paper. The weighted knotting and linking pattern of the oriented link L is the weighted linking pattern along with the weight |a2(Li)| on the vertex corresponding to component Li, for each i. We use J∇L for the closure of the symmetric difference and J∇L is J∇L if = 1 and J∇∅ = J, when = 0. The proof of the following three lemmas is virtually identical to those given in [2] and we refer the reader there for details. The only novelty is in the proof of the first lemma where, taking advantage of the 2- directionality of Bi, we choose vertices xi and yj in Bi so that both paths in Bi are directed from xi to yj. 3 Lemma 5. Let λ > 0. In an embedding of DKr in R , let A1, ··· ,An be disjoint 1-directional cycles and B1, ··· ,B6n+6 disjoint 2-directional cycles such that lk(Ah,Bi) ≥ λ for all h and i. Then there exist disjoint 0 2-directional cycles C1,C2,C3,C4 ∈ {Bi} and a 1-directional cycle W S 0 in DKr with verticies on i Bi such that W intersects each Ci in 0 exactly one arc. In addition, |lk(Ah,W ∇1C1∇2C2∇3C3∇4C4)| ≥ λ for every h and every choice of 1, ··· , 4 ∈ {0, 1}. 3 Lemma 6. Let λ > 0. In an embedding of DKr in R , let A1, ··· ,An be disjoint 1-directional cycles and let B1, ··· ,B6n+6 be disjoint 2- directional cycles such that lk(Ah,Bi) ≥ λ and |lk(Bi,Bj)| ≥ λ for all h, i, and j. Then there exists a 1-directional cycle K in DKr with S 2 verticies on iBi such that |a2(K)| ≥ λ /16 and |lk(Ah,K)| ≥ λ for every h.

Lemma 7. Let n, λ ∈ N. Suppose that a complete graph DKr embedded 3 n in R contains a link L0 with f (n) 2-directional components, where LINKING AND KNOTTING IN DIGRAPHS 9 f(n) = n − 1 + (6n)2n−2, such that the linking number of every pair of components of L0 has absolute value at least λ. Then DKr contains a link with 1-directional components Q1,...,Qn such that for every 2 i 6= j, |lk(Qi,Qj)| ≥ λ and |a2(Qi)| ≥ λ /16.

Proof. (of Theorem 2) As usual,√ the proof is very similar to that given in [2]. Let λ = Max{α, 4 α}, let f(n) = n − 1 + (6n)2n−2, and n let m = f (n). By Theorem 1, there exists a graph DKr such that 3 every embedding of DKr in R contains a link L0 with m δ-directed components such that each pair of components has a linking number whose absolute value is at least λ. Let δ = 2. By Lemma 7, every 3 embedding of DKr in R contains a link with 1-directional components Q1,...,Qn where, for every i 6= j, |lk(Qi,Qj)| ≥ λ ≥ α and |a2(Qi)| ≥ λ2/16 ≥ α. References [1] J. Conway and C. Gordon, Knots and links in spatial graphs, J. of Graph Theory, 7 (1983) 445–453. [2] E. Flapan, B. Mellor, and R. Naimi, Intrinsic Linking and Knotting are Arbi- trarily Complex, Fund. Math, 201 (2008) 131-148. [3] H. Sachs, On spatial representations of finite graphs, Finite and infinite sets, Vol. I, II (Eger, 1981), 649–662, Colloq. Math. Soc. JánosBolyai, 37, North- Holland, Amsterdam, 1984. [4] T. Fleming and J. Foisy, Intrinsically knotted and 4-linked directed graphs, J. Knot Theory Ramifications 27 (2018), 1850037, 18 pp.

Department of Mathematics and Statistics, California State Uni- versity, Chico, Chico, CA 95929-0525 E-mail address: [email protected]

Department of Mathematics, Occidental College, Los Angeles, CA 90041 E-mail address: [email protected] E-mail address: [email protected]